3D Electron Fluid Turbulence at Nanoscales in Dense Plasmas

We have performed three dimensional nonlinear fluid simulations of electron fluid turbulence at nanoscales in an unmagnetized warm dense plasma in which mode coupling between wave function and electrostatic potential associated with underlying electron plasma oscillations (EPOs) lead to nonlinear cascades in inertial range. While the wave function cascades towards smaller length scales, electrostatic potential follows an inverse cascade. We find from our simulations that quantum diffraction effect associated with a Bohm potential plays a critical role in determining the inertial range turbulent spectrum and the subsequent transport level exhibited by the 3D EPOs.


I. INTRODUCTION
Studies of collective phenomena at nanoscales in dense matters are of great importance in diverse areas of physics, including the fields of plasmonics [1,2,3,4,5], semiconductors [6], nano-electromechanical systems [7], quantum-diodes [8], nanotubes and nanowires [9], quantum free electron lasers [10], as well as astrophysical bodies [11,12] and intense laser-solid density plasma interaction experiments [13] for x-ray and γ-ray sources. In dense plasmas, the electrons are highly degenerate and quantum mechanical effects (e.g. electron tunneling arising from the finite width of the electron wave function) play an essential role at nanoscales. Since degenerate electrons follow the Fermi-Dirac statistics, there appear new electron equation of state and new forces involving the quantum Bohm potential [14] and electron-1/2 spin effects [15] in dense quantum plasmas. It then turns out that due to the intrinsic nonlinearities associated with the Fermi pressure law and quantum forces, there exists possibility of localizing electrostatic [16,17] and electromagnetic [18] wave energies at nanoscales in dense quantum plasmas. Here we report simulation studies of threedimensional (3D) electron fluid turbulence at nanoscales in an unmagnetized warm dense plasma. It is found that 3D nonlinearly interacting electron plasma oscillations (EPOs) [19] in a dense quantum plasma exhibit nano-structures and associated energy spectra that are markedly different from those reported earlier for the 2D case [17]. Furthermore, we stress that the present 3D turbulence properties of electron plasma oscillations in our dense quantum plasma are significantly different from those in a classical plasma [20,21,22]. In the latter, strong electron plasma wave turbulence has been studied by invoking parametric interactions [23] and by using either multi-dimensional cubic nonlinear Schrödinger equation [21,22] or Zakharov equations [20], which are different from our 3D coupled nonlinear Schrödinger-Poisson equations in dense quantum plasmas.
The surge for studying numerous nonlinear processes in dense quantum plasmas lies in a hope to transfer information through localized nano-structures one is able to create and sustain in plasmas. The present work dealing with 3D electron fluid turbulence shares a great deal of knowledge with classical fluid turbulence [24,25], plasma turbulence [26,27], and superfluid turbulence involving the Bose-Einstein condensates (BECs) [28,29] in ultracold gases. Both in fluids and plasmas as well as in BECs, one encounters the phenomena of inverse energy cascades in which energy transfer from small scales sustains large scale circulations/structures in the flow, and results in a steady-state inertial range with powerlaw scaling, as was originally predicted by Kolmogorov, Kraichnan and Iroshnik [24,25].
while the dynamical equations depicting inverse cascades in fluids and plasmas are the Navier-Stokes and Charney-Hasegawa-Mima equations [26,27], the energy cascade scenario in BECS is described by the Gross-Pitaevskii equation [28,29]. In Sec II, we describe model equation governing the dynamical evolution of 3D dense fermi quantum plasma. We also present conservation laws admitted by the set of 3D equations. In Sec III, nonlinear 3D simulation results describing turbulence in such system are described. Mode structures and corresponding Kolmogorov-like spectra are also discussed. Turbulence transport is described in Sec IV, and finally the conclusions are contained in Sec V.

II. MODEL EQUATIONS
In dense quantum plasmas, the Wigner-Poisson (WP) model has been used to derive a set of quantum hydrodynamic (QHD) equations [30] in the mean field approximation. The QHD In this paper, we carry out simulations of 3D NLS and Poisson equations in order to understand the properties of 3D electron fluid turbulence (involving nano-structures and associated electron transport) in a warm dense plasma. We find that nonlinear couplings be-tween different scales EPOs are responsible for creating small-scale electron density clumps, while the electrostatic potential assumes large-scale structures. The total energy associated with our 3D electron fluid turbulence at nanoscales processes a characteristic spectrum which is a non-Kolmogorov-like.
For our 3D electron fluid turbulence studies, we use the NLS-Poisson equations [16,30] i and which are valid at zero electron temperature for the Fermi-Dirac equilibrium distribution.
In Eqs. (1) and (2) the wave function Ψ is normalized by √ n 0 , the electrostatic potential ϕ by k B T F /e, the time t by the electron plasma period Ω −1 pe , and the space r by the Fermi Debye radius λ D . We have introduced the notations 0 , e is magnitude of the electron charge, and Ω pe = (n 0 e 2 /ǫ 0 m e ) 1/2 is the unperturbed electron plasma frequency. The origin of the various terms in Eq.(1) is obvious. The first term is due to the electron inertia, the H-term is associated from the quantum diffraction effect involving the Bohm potential, ϕΨ comes from the nonlinear coupling between the scalar potential (associated with the space charge electric field resulting form oscillations of the electrons around immobile ions) and the electron wave function, and the fourth term in the left-hand side of (1) is the contribution of the 3D electron pressure (p e = m e V 2 F n 5/3 e /5n  1) and (2). We note that linearizing the latter one obtains the EPO frequency ω = Ω 2 , which exhibits the dispersive behavior of the EPOs. In the short wavelength regime characterized by k 2 ≫ 4m 2 e V 2 F /h 2 , one notices that the dispersion associated with electron tunneling effect dominates over that involving the quantum statistical electron pressure.

III. NONLINEAR 3D SIMULATIONS OF QUANTUM PLASMAS
We have developed 3D fluid code to investigate nonlinear interactions between multiscales EPOs described by (1) and (2). Our 3D fluid code is based on Fourier expansion of the bases using a fully de-aliased pseudospectral numerical scheme [31]. The nonlinear de-convolution of Fourier modes is performed by computing the nonlinear triad interactions  1) and (2), as presented above. The temporal integration is performed by 4th order Runge Kutta method. The spectral distribution for turbulent fluctuations is initialized isotropically (no mean fields are assumed) with random phases and amplitudes in Fourier space.
The evolution variables use periodic boundary conditions. The initial isotropic turbulent spectrum was chosen close to k −2 , with random phases in all three directions. The choice of such (or even a flatter than −2) spectrum treats the turbulent fluctuations on an equal footing and avoids any influence on the dynamical evolution that may be due to the initial spectral non-symmetry. Note, however, that the local as well as global mean flows may subsequently be generated by self-consistently excited nonlinear instabilities.  The localized initial turbulent spectral distribution, concentrated at the lower wavenumbers, evolves in time following 3D nonlinear electron plasma wave interactions. Since the initial energy is localized in the large scale fluctuations, the latter drive turbulent processes through migration of energy towards relatively small scales Consequently, larger eddies transfer their energy to smaller ones through a forward cascade. During the forward cascade process, each Fourier mode in the inertial range spectrum obeys the vector triad constraints [23] imposed by the vector relation k + p = q. These nonlinear interactions involve the neighboring Fourier components (k, p, q) that are excited in the local inertial range turbulence. We have performed a number of simulations to verify the consistency of our results in a strong turbulence regime. In our 3D simulations, we have explored two dense plasma systems that are characterized by different physical parameters, viz. dense warm plasmas in the next generation laser-based plasma compression (LBPC) schemes [32], and the superdense astrophysical bodies [33](e.g. interior of white dwarf stars). It is expected that in LBPC schemes, the electron number density may reach 10 33  and an inverse cascade of magnetic helicity. In these processes, the energy cascades towards smaller length-scales, while the magnetic helicity in MHD transfers spectral power towards larger length-scales. By contrast, the fluid vorticity in 3D hydrodynamics is prohibited from an inverse cascade. The randomly excited 3D Fourier modes nonetheless transfer the spectral energy by conserving the constants of motion in k-space. In freely decaying quantum electron fluid turbulence reported here, the energy contained in the large-scale eddies is transferred to the smaller scales, leading to a statistically stationary inertial regime associated with the forward cascades of one of the invariants. Decaying turbulence often leads to the formation of coherent structures as turbulence relaxes, thus making nonlinear interactions rather inefficient when they are saturated. It is to be noted further that the long scale flow generation in our 3D simulations is observed to be directly proportional to the parameter H. Intermittent flows are thus generated for a small value of H, while strong and large scale flows in the ES potential are formed when the magnitude of H is large (see, e.g. Fig. (1)). The physical basis of such observation can be elucidated from the following arguments. The parameter H, which is the ratio between the energy density of the EPOs and the electron kinetic energy density of a warm dense quantum plasma, is associated with a diffraction-like term in Eq. (1) i.e. H∇ 2 Ψ. In this term, the negative imaginary part of the complex evolutionary variable Ψ essentially determines the rate of dissipation corresponding to the smaller scales. The smaller is H, more the dissipation is concentrated at the smaller scales and vice versa. For a moderately higher magnitude of the H parameter, there exists a strong tendency in EPO's to dissipate the smaller and intermittent turbulent eddies. It is therefore this H parameter which essentially characterizes electron flows at nanoscales in our 3D simulations.
While the power spectrum for nonlinear EPOs exhibits an interesting feature in our 3D simulations, its scaling is not universal and is determined critically by the parameter H. For instance, we find a 3D Kolmogorov-like power spectrum k −11/3 in some range of H values as shown in Fig. (2). The corresponding omnidirectional spectrum thus exhibits a k −5/3 scaling. Spectral index nevertheless changes with H, as noted also in the study of 2D fluid turbulence [17]. However, the spectral slope in the latter was found to be close to the Iroshnikov-Kraichnan (IK) power law [34,35] k −3/2 , rather than the usual Kolomogrov scaling [36] k −5/3 . The origin of the differences in the observed spectral indices resides with the nonlinear character of the underlying warm dense plasmas, as nonlinear interactions in 2D and 3D systems are governed typically by different nonlinear forces. The latter modify the spectral evolution of turbulent cascades to a significant degree.

IV. ELECTRON TRANSPORT CAUSED BY TURBULENT FIELDS
We finally study the electron diffusion coefficient in the presence of small and large scale turbulent EPOs. The effective electron diffusion coefficient produced by the momentum transfer can be calculated from D ef f = ∞ 0 P(r, t) · P(r, t + t ′ ) dt ′ , where P is the electron momentum and the angular bracket denotes spatial averages and the ensemble averages are normalized to unit mass. The effective diffusion coefficient D ef f , resulting from 3D structures, essentially relates the diffusion processes associated with random translational motions of electrons in nonlinear fields of localized EPOs. It is remarkable to note that the electron transport can be effectively suppressed when the magnitude of the parameter H is decreased. This is shown in Fig. (3). The outcome of Fig. (3) is contrary to our 2D results [17], where the effective electron diffusion was lower when the field perturbations were Gaussian and increases later rapidly with the eventual formation of longer length-scale structures. Unlike the 2D case, the electron diffusion in 3D case is suppressed eventually because of the presence of small scale turbulent fluctuations. However, the steady state diffusion coefficient increases with the increase of H. This can be understood as follows. Higher the H valuer, the stronger is the small scale damping. When the small scale fluctuations are smeared out, the dynamical evolution of 3D quantum electron plasma wave turbulence is predominantly governed by the large scale flows, which consequently lead to a higher level of transport. This finding is further consistent with the fact that the parameter H dictates the characteristic evolution properties of the 3D EPOs, as described above.

V. SUMMARY
In conclusion, we have presented results from 3D nonlinear fluid simulations of the electron fluid turbulence at nanoscales in an unmagnetized warm dense plasma. The mode couplings between the electron wave function and the electrostatic potential associated with the underlying electron plasma oscillations (EPOs) lead to the onset of nonlinear interactions and subsequent cascades in the inertial range. We find from our 3D simulations that the dispersion effect associated with the quantum Bohm potential plays a critical role in determining the inertial range turbulent spectrum and the subsequent transport level exhibited by the 3D EPOs. For instance, a Kolmogorov-like k −11/3 3D spectrum is observed for H = 0.4, whereas the spectrum flattens out for a smaller value of H. Correspondingly, the electron transport is higher for the higher H values. Finally, the wave function cascades towards smaller length scales, while the electrostatic potential follows an inverse cascade.
We reiterate that the present investigation of 3D EPO turbulence is a necessary prerequisite